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We describe the multiplicative structures that arise on categories ofequivariant modules over certain equivariant commutative ring spectra. Buildingon our previous work on N-infinity ring spectra, we construct categories ofequivariant operadic modules over N-infinity rings that are structured byequivariant linear isometries operads. These categories of modules are endowedwith equivariant symmetric monoidal structures, which amounts to the structureof an "incomplete Mackey functor in homotopical categories". In particular, weconstruct internal norms which satisfy the double coset formula. We regard thework of this paper as a first step towards equivariant derived algebraicgeometry.

G-symmetric monoidal categories of modules over equivariant commutative ring spectra